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Theorem dfon2lem1 35784
Description: Lemma for dfon2 35793. (Contributed by Scott Fenton, 28-Feb-2011.)
Assertion
Ref Expression
dfon2lem1 Tr {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)}

Proof of Theorem dfon2lem1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 truni 5275 . 2 (∀𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)}Tr 𝑦 → Tr {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)})
2 nfsbc1v 3808 . . . . 5 𝑥[𝑦 / 𝑥]𝜑
3 nfv 1914 . . . . 5 𝑥Tr 𝑦
4 nfsbc1v 3808 . . . . 5 𝑥[𝑦 / 𝑥]𝜓
52, 3, 4nf3an 1901 . . . 4 𝑥([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦[𝑦 / 𝑥]𝜓)
6 vex 3484 . . . 4 𝑦 ∈ V
7 sbceq1a 3799 . . . . 5 (𝑥 = 𝑦 → (𝜑[𝑦 / 𝑥]𝜑))
8 treq 5267 . . . . 5 (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
9 sbceq1a 3799 . . . . 5 (𝑥 = 𝑦 → (𝜓[𝑦 / 𝑥]𝜓))
107, 8, 93anbi123d 1438 . . . 4 (𝑥 = 𝑦 → ((𝜑 ∧ Tr 𝑥𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦[𝑦 / 𝑥]𝜓)))
115, 6, 10elabf 3675 . . 3 (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)} ↔ ([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦[𝑦 / 𝑥]𝜓))
1211simp2bi 1147 . 2 (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)} → Tr 𝑦)
131, 12mprg 3067 1 Tr {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)}
Colors of variables: wff setvar class
Syntax hints:  w3a 1087  wcel 2108  {cab 2714  [wsbc 3788   cuni 4907  Tr wtr 5259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-v 3482  df-sbc 3789  df-ss 3968  df-uni 4908  df-iun 4993  df-tr 5260
This theorem is referenced by:  dfon2lem3  35786  dfon2lem7  35790
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